Graphing Quadratic Functions
MA.912.A.7.1 Graph quadratic equations.
MA.912.A.7.6 Identify the axis of symmetry,
vertex, domain, range, and intercept(s) for a
given parabola
Quadratic Function
y = ax2 + bx + c
Quadratic Term
Linear Term
Constant Term
2
What is the linear term of y = 4x – 3? 0x
2
What is the linear term of y = x - 5x ? -5x
2
What is the constant term of y = x – 5x? 0
Can the quadratic term be zero?
No!
Quadratic Functions
parabola
The graph of a quadratic function is a:
y
A parabola can open up
or down.
Vertex
If the parabola opens up,
the lowest point is called
the vertex (minimum).
If the parabola opens down,
the vertex is the highest point
(maximum).
Vertex
NOTE: if the parabola opens left or right it is not a function!
x
Standard Form
The standard form of a quadratic function is:
y = ax2 + bx + c
y
The parabola will
open up when the a
value is positive.
a>0
The parabola will
open down when the
a value is negative.
a¹0
x
a<0
Axis of Symmetry
Parabolas are symmetric.
If we drew a line down
the middle of the
parabola, we could fold
the parabola in half.
Axisy of
Symmetry
We call this line the
Axis of symmetry.
x
If we graph one side of
the parabola, we could
REFLECT it over the
Axis of symmetry to
graph the other side.
The Axis of symmetry ALWAYS
passes through the vertex.
Finding the Axis of Symmetry
When a quadratic function is in standard form
y = ax2 + bx + c,

b
the equation of the Axis of symmetry is x 
2a
This is best read as …
‘the opposite of b divided by the quantity of 2 times
a.’
2
Find the Axis of symmetry for y = 3x – 18x + 7
a = 3 b = -18
18
18
x

2 3
6
3
The Axis of
symmetry is
x = 3.
Finding the Vertex
The Axis of symmetry always goes through the
Vertex Thus, the Axis of symmetry gives
_______.
X-coordinate of the vertex.
us the ____________
Find the vertex of
y = -2x2 + 8x - 3
STEP 1: Find the Axis of symmetry
b
x
2a
a = -2
b=8
-8
-8
x=
=
= 2
2(-2)
-4
The xcoordinate
of the vertex
is 2
Finding the Vertex
Find the vertex of
y = -2x2 + 8x - 3
STEP 1: Find the Axis of symmetry
b
8
8
x 

2
2a 2(2) 4
STEP 2: Substitute the x – value into the original equation
to find the y –coordinate of the vertex.
y = -2 ( 2 ) +8 ( 2 ) - 3
2
= - 2 ( 4 ) + 16 - 3
= - 8 + 16 - 3
= 5
The
vertex is
(2 , 5)
Graphing a Quadratic Function
There are 3 steps to graphing a parabola in
standard form.
b
STEP 1: Find the Axis of symmetry using: x 
2a
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across
the Axis of symmetry. Then connect the five points
with a smooth curve.
MAKE A TABLE
using x – values close to
the Axis of symmetry.
Graphing a Quadratic Function
Graph : y = 2x - 4x -1
2
y
x =1
STEP 1: Find the Axis of
symmetry
-b
4
x=
=
=1
2a 2(2)
STEP 2: Find the vertex
Substitute in x = 1 to
find the y – value of the
vertex.
y = 2(1)2 - 4(1)- 1 = - 3
x
Vertex : (1, - 3)
Graphing a Quadratic Function
Graph : y = 2x - 4x -1
2
STEP 3: Find two other
points and reflect them
across the Axis of
symmetry. Then connect
the five points with a
smooth curve.
x
y
2 –1
3
5
y = 2(2)2 - 4(2)- 1 = - 1
y = 2(3)2 - 4(3)- 1 = 5
y
x
Y-intercept of a Quadratic Function
y = 2x - 4x -1
2
Y-axis
y
The y-intercept of a
Quadratic function can
Be found when x = 0.
y = 2x 2 - 4 x - 1
= 2(0) - 4(0) - 1
2
= 0 - 0 -1
= -1
The constant term is always the y- intercept
x
Solving a Quadratic
The x-intercepts (when y = 0) of a quadratic function
are the solutions to the related quadratic equation.
The number of real solutions is at
most two.
No solutions
One solution
Two solutions
X=3
X= -2 or X = 2
Identifying Solutions
Find the solutions of 2x - x2 = 0
The solutions of this
quadratic equation can be
found by looking at the
graph of f(x) = 2x – x2
The xintercepts(or
Zero’s) of
f(x)= 2x – x2
are the solutions
to 2x - x2 = 0
X = 0 or X = 2